Weingarten — who was head of the United Federation of Teachers in New York before becoming chief of the national American Federation of Teachers — is against charter schools, teacher evaluations and most of the other reforms Mayor Bloomberg enacted.

“She wants the job, and de Blasio’s people have been making calls, asking about Weingarten and testing the reaction,” said one well-placed source in the public education sector.

“The idea of putting a union chief in charge of a school system is mind-boggling,” said a political consultant. “It strains credulity that de Blasio would go that far.”